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## G = C62.116C23order 288 = 25·32

### 111st non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.116C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.116C23
 Lower central C32 — C3×C6 — C62.116C23
 Upper central C1 — C22 — C23

Generators and relations for C62.116C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=b3d >

Subgroups: 1538 in 331 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×8], C3 [×2], C3, C4 [×4], C22, C22 [×2], C22 [×20], S3 [×20], C6 [×6], C6 [×11], C2×C4 [×8], C23, C23 [×10], C32, Dic3 [×4], C12 [×4], D6 [×64], C2×C6 [×6], C2×C6 [×11], C22⋊C4 [×4], C22×C4 [×2], C24, C3⋊S3 [×4], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×4], C2×C12 [×4], C22×S3 [×34], C22×C6 [×2], C22×C6, C2×C22⋊C4, C3×Dic3 [×4], C2×C3⋊S3 [×8], C2×C3⋊S3 [×10], C62, C62 [×2], C62 [×2], D6⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], S3×C2×C4 [×4], S3×C23 [×3], C6.D6 [×4], C6×Dic3 [×4], C22×C3⋊S3 [×2], C22×C3⋊S3 [×4], C22×C3⋊S3 [×4], C2×C62, S3×C22⋊C4 [×2], C6.D12 [×2], C3×C6.D4 [×2], C2×C6.D6 [×2], C23×C3⋊S3, C62.116C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×4], C23, D6 [×6], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×4], C22×S3 [×2], C2×C22⋊C4, S32, S3×C2×C4 [×2], S3×D4 [×4], C6.D6 [×2], C2×S32, S3×C22⋊C4 [×2], C2×C6.D6, Dic3⋊D6 [×2], C62.116C23

Permutation representations of C62.116C23
On 24 points - transitive group 24T673
Generators in S24
```(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 17 5 15 3 13)(2 18 6 16 4 14)(7 19 9 21 11 23)(8 20 10 22 12 24)
(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)
(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)```

`G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,19,9,21,11,23)(8,20,10,22,12,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20)>;`

`G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,19,9,21,11,23)(8,20,10,22,12,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20) );`

`G=PermutationGroup([(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,17,5,15,3,13),(2,18,6,16,4,14),(7,19,9,21,11,23),(8,20,10,22,12,24)], [(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,22,16,19),(14,23,17,20),(15,24,18,21)], [(1,21,4,24),(2,20,5,23),(3,19,6,22),(7,18,10,15),(8,17,11,14),(9,16,12,13)], [(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)])`

`G:=TransitiveGroup(24,673);`

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3A 3B 3C 4A ··· 4H 6A ··· 6F 6G ··· 6Q 12A ··· 12H order 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 9 9 9 9 18 18 2 2 4 6 ··· 6 2 ··· 2 4 ··· 4 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4×S3 S32 S3×D4 C6.D6 C2×S32 Dic3⋊D6 kernel C62.116C23 C6.D12 C3×C6.D4 C2×C6.D6 C23×C3⋊S3 C22×C3⋊S3 C6.D4 C2×C3⋊S3 C2×Dic3 C22×C6 C2×C6 C23 C6 C22 C22 C2 # reps 1 2 2 2 1 8 2 4 4 2 8 1 4 2 1 4

Matrix representation of C62.116C23 in GL6(𝔽13)

 1 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 10 0 0 0 0 5 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 12 3 0 0 0 0 8 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 5 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,5,0,0,0,0,10,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,12,8,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,5,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C62.116C23 in GAP, Magma, Sage, TeX

`C_6^2._{116}C_2^3`
`% in TeX`

`G:=Group("C6^2.116C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,622);`
`// by ID`

`G=gap.SmallGroup(288,622);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,422,219,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^3*c,e*d*e=b^3*d>;`
`// generators/relations`

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